A note on the capacity of the binary perceptron
This is an incremental theoretical contribution to statistical physics and machine learning, addressing a specific mathematical problem with limited direct application.
The paper tackles the long-standing problem of determining the capacity of the binary perceptron, providing a complete proof that the capacity is less than 0.847, which refines previous upper bounds.
Determining the capacity $α_c$ of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of $α_c$, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that $α_c$ < .996, while Krauth and Mezard outlined an argument which can be used to show that $α_c$ < .847. The purpose of this expository note is to record a complete proof of the bound $α_c$ < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron